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## Homework Statement

Find a fundamental set of real solutions of the system.

x'=[-0.5 1 ]x

[-1 -0.5]

## The Attempt at a Solution

I calculated the eigenvalues to be r1 = -0.5+i and r2 = -0.5-i

Then, attempting to calculate the eigenvectors,

I plugged the numbers into the system

[-0.5+0.5-i 1][v1] = [0]

[-1 -0.5+0.5-i ][v2] [0]

which simplifies to be

[-i 1] [v1] = [0]

[-1 -i ][v2] [0]

afer matrix multiplication, I get two formulas

-1(v1)i+(v2)=0

-1(v1)-(v2)i=0

However, I am at a loss for the next step. Usually when I am finding the eigenvector those two equations will have the same ratio. Please advise.